Problem: As $n$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $11n+3$ and $6n+1$?
Explanation: We use the Euclidean Algorithm. \begin{align*}
\gcd(11n+3, 6n+1) &= \gcd(6n+1, (11n+3) - (6n+1)) \\
&= \gcd(6n+1, 5n+2) \\
&= \gcd(5n+2, (6n+1)-(5n+2)) \\
&= \gcd(5n+2, n-1) \\
&= \gcd(n-1, (5n+2)-5(n-1)) \\
&= \gcd(n-1, 7).
\end{align*}Therefore, if $n-1$ is a multiple of 7, then the greatest common divisor of $11n+3$ and $6n+1$ is 7. Otherwise, the greatest common divisor is 1. This implies that the maximum possible value for the greatest common divisor of $11n+3$ and $6n+1$ is $\boxed{7}$.